Adjusting Mercury Concentration for Fish-Size Covariation: A tivariate A ternative to Bivariate Regression

Keith M. Sornersl and Donald A. Jackson Department ofdooIogy, University of Toronto, Toronto, ON A455 I / % 1 , Canada

Sorners, K.M., and D.A. Jackson. 1993. Adjusting mercury concentration for fish-size covariation: a multivariate alternative to bivariate regression. Can. J. Fish. Aquat. Sci. 50: 2388-2396.

Regression-based methods like analysis of covariance (ANCOVA) are frequently used to adjust one variable for the correlated influence of a second less interesting variable (e.g., mercury concentration and fish size). However, the influence af the covariate (i.e., fish size) is not removed unequivocaily when regression slopes are not parallel. Using data on tissue-mercury concentration and Fish size from 30 populations of lake trout (Sa/s/e/inus namaycushb, we show that data adjusted to a common size with bivariate regression can retain information associated with the original sine differences. As an alternative, we use univariate and bivariate summary statistics from each population as raw data in a multivariate anaiysis to search for differences among populations. Ordination axes resulting from this analysis exhibited both small- and large-scale spatial autocarrelation. Localized spatial patterns probably reflect similar geochemical features of the watersheds of neigh bouring lakes in small geographic areas. In contrast, regional spatial autocorrelation suggested broad-scale patterns that may implicate atmospheric inputs s f mercury. As an extension of this multivariate approach, both regional and local patterns could be compared with environmental variables to reveal correlations that naay suggest new cause-and-effect hypotheses.

Des methodes fondees s~ i r la rkgressian comme I'ana[yse de la covariance sont souvent eetilis6es pour ajuster une variable pour tenir compte de I'influence correlee d'une deuxieme variable mains interessante (p.ex., la concentration de mercure et la taille du poisson). Toutefois, I'influence de %a covariable (c.-2i.-d !a taille du poisson) n'est pas supprimke de f a ~ o n univoque lorsque les pentes de regression ne sont pas parall5les. A I'aide de donnkes sur la concentration de naercure dans ies tissus et la tailie des psissons de 30 populations de touladi (Salvelinus namaycush), nous avons montre que les donnees ajustees a une tarlle courante au rnoyen de la regression 2 deuw variables peuvent garder des informations assaciees aux differences de taille originales. Cornme autre psssibilite, nous utilisons les statistiques sommaires a une variable et & deux variables de chaque population cornme donnees brutes dans le cadre d'une analyse a plusieurs variables afin de deceler des differences entre les populations. Les axes d'ordination obtenues 2 partir de cette analyse demontraient une autocorr6latian spatiale sur une petite 6chelle et sur une grande echelle. Des configurations spatiales localiskes refletent probablernent des caracteres geochimiques similaires de bassins hydrographiques des lacs voisins dans de petites nones gkographiy ues. Par contraste, une autocorr6lation spatiale rkgionale semblait indiquer des configurations sur une vaste kchelle susceptibles de comporter des apports atmosphkriques de mercure. Pour 6Iargir cette approehe 3 plusieurs variables, des configurations regionales et locales pourraient &re comparees a des variables environnementales afin de mettre & jour des correlations qui peuvent indiquer de nouvelles hypotheses cause-effet.

Received July 24, 9999 7 Accepted May 3 9 , 1993 (JB658)

'he proper method to correct or adjust one variable for relatively uninteresting variation in a second variable is a long-standing issue ian biological studies (e.g . , Atchle y

1978; Smith 1984; Reist 1985, 1986; Packard and Boardman 1988). The use of indices or ratios was once widely accepted as one such adjustment (e.g., condition factor, gsnadosornatic index (GSI); liver somatic index (ESI), and so on; see Leatherland md Sonstegad 1987; Munkittrick and Dixon 1988; Willis 1989). But the recognition of statistical problems with ratios (e.g., Pearson B 897; Chayes 1971 ; Atchley et al. 1976; Atchley and Anderson 1978; Renney 1982: Phillips 1 983; Gibson 1984; Packard and Boardman 1 987,1988) has prompted the evaluation and use of numerous alternatives (e.g., Cesmccini

'Present address: Limnslogy Section, Water Resources Branch, Ontario Ministry of the Environment, P.O. Box 39, Dorset, ON POA IEO, Canada.

1977; Strauss 1985; Craig and BabaBuk 1989; Cone 1989; dos Reis et al. 1990; Jackson et al. 1990; Murphy et al. 1998).

Many of these alternatives incorporate regression-based procedures (e.g., Atchley 1978; Smith 1984; Green 1986); however, interpretive problems arise when co~~~parimg pspu%ations if regression slopes difTer (Thsrpe 1976; Weist 1985). Frequently, such problems emerge in studies of contaminants in animal tissues where contaminant concentration increases with animal size (e.g., mercury in fish; Wren and MacCrirnmon 1983; Wada et al. 1986; McMux-try et al. 1989; Wiener et al. 1990; Wren et al. 199 1 ) or with lipid concentration (e.g., Roberts et al. 1977; Borgrnann and Whittle 1983; M6ller et al. 1983). At issue is how to remove the correlated influence of animal size or lipid concentration so differences in contaminant concentrations can be examined and better unders tosd.

One common approach employs analysis of covariance (ANCOVA) to compare slopes and intercepts of several regres-

2388 Can. 9. Fish. Aquar. Sci., 141. 50,1993

NORTHWESTERN ONTARIO

FlSH LENGTH (em)

FIG. 1. Regressions of mercury concentration on fish length for 10 populations of Iake trout froin rlorthwestern Ontario lakes. The adjusted mercury concentration for a standardized length of 44 cm is indicated (by the broken vertical line) for each population based on (A) individual regressions and (B) common within-groups regressions. Line lengths span the range of lengths in the collection for each population.

sisns, thereby using bivariate regression to ad-just for uncon- trolled variation in the independent variable (is., A') (e.g., Green 1986; Packard and Boardman 1987, 1988; Tracy and Sugar 1989). An ANCOVA is relatively simple with a small number of regressions, but with large data sets and many regressions, the revelation that slopes differ leaves much to be desired. DifTeri~ng s lops confound further analysis, much like a significant inter- action between main effects in an analysis sf variance (ANOVA) (see Sokal and Rshlf 1981, p. 414-417). As a result, the ANCOVA will not reliably evaluate whether intercepts differ (i.e., the usual test of interest) when regressions are not parallel. This limitation is particularly frustrating when the ccsvarying X variable could not be "controlled" at the time of data collection (e.g., Reist 1985; Packard and Boardman 1987; Meffe et d. 1988).

The problem of size adjustment only materializes when slopes are not parallel because ANCOVA automatically adjusts for the covariate and compares intercepts (or adjusted means) when the slopes are not different. However, when slopes differ, the common within-groups slope is recommended as the best estimate of the common slope (Thorpe 1976; Thorpe and Lemy 1983; Reist 2985) (e.g., see Bdtz and Moyle 198 1; Hhssen et d. 19811; Reist and Crossman 1987). This estimated slope is then used to calculate the intercept and adjusted mean concentation for each population, or the predicted concentration at some arbitrarily selected standard size. However, Reist (1985) cautioned that results from this type of adjustment may vary depending on the particular data set k ing malysed (e.g., see m o p e 1976; Rayner 1985; Shea 1985).

In essence, the within-groups approach ignores the fact that slopes differ significantly among populations and simply

Con. J . Fish. Aquai. Sci., Vol. 50.1993

estimates a common slope across all populations (see Fig. 1B). The suitability of this estimate depends on the relative variation of each population, since varying populations can have undue influence on the estimate of the common slope (e.g., see Thorpe 1976; Pimentel 1979, p. 81-91; Thsrpe and Lemy 1983).

An alternative approach uses the individuaj regression equations to estimate mercury concentration at a common size for each population (see Fig. 1 A) (e.g., Taylor and McPhailE985; Johnson 198% McMuflry et al. 1989). By using individual equations, significant differences in slopes are ignored although these differences are incorporated into the standardization. As a result, his method may combine differences in slopes with differences in intercepts (see below). The relative contributions of the slope and intercept depend on the scale of the original axes (e.g., the size sf the slope relative to the intercept) and the common size selected. Because the choice to use the common, within-groups approach may also affect the results, we must choose between two potential distortions in order to examine other factors that correlate with observed differences among populations.

McMaaary et al. (1989) provided a recent example in an examination of the concentrations of mercury in dorsal muscle sf populations of lake trout (Salve%irzus nanzaysush) md smdtllmsuth bass (Mkcrcppterus hlsrnieu) from approximately 964 lakes in Ontario. M.J. McMurtry (personal communication) first used ANCOVA to compare regressions of logarithmically transformed mercury concentration on fish length for each population because mercury concentration in fish muscle tends to csvary with fish size (e.g., Scott and Amstrong 1972; Scott 1974; MacCrimmon et all. 1983; Sprenger et al. 1988). Udor- tunately? the regression slopes differed among populations (e.g.,

see Fig. 1A). To explore environmental factors that correlated with these differences, McMurtry et al. (1989) subsequently chose to adjust observed mercury concentrations to a standard lake trout size of 44 cm using individual regression equations for each population (e.g., Fig. 1 A). These adjusted values were then cot-related with physical and chemical variables from each lake to determine which environmental factors best predicted size-adjusted tissue-mercury concentrations.

The estimated mercury concentration used by McMurtry et al. (1989) is a composite index based on the observed regression intercept plus the slope multiplied by the adjusted size (compare Fig. BA and BB). Although means and standard deviations of the original variables (i.e., fish length and mercury concentration) differed among populations, these differences were ignored because differences in the size distributions of the samples likely reflected collecting biases ( M . McMurtry, personal communication).

By using either the within-groups or the individual regression adjustment, the bivariate relationships are reduced to a single index for eachpopulation (see Fig. 1). As a result, the means and standard deviations for each variable are combined with the regression slopes, intercepts, and correlations (i.e., weighted by the correlation; see Jensen 1986) to estimate a mercury con- centration that should be independent of fish length. Unfortunately. this data-reduction approach pools infomation inconsistently because different weights are used for each population. As a result, similar size-adjusted values may wise even though the original values differed considerably. Hence- forth, this difference is lost from subsequent analyses.

Alternatively, an analysis using univariate and biivariate summary statistics as new variables for each population may provide insight with less infomation loss (e.g., Atchley and Rutledge 1980; Rohlf and Archie 1984; Ferson et d. 1985). Because the summary statistics me estimated from the same variables, these statistics are not independent and hence, a multivariate analysis is warranted. If a priari groups or classes sf populations exist (e.g., geographic regions), a multivariate analysis sf vandance (i.e., MANOVA) might be employed. Otheswise data exploration techniques such as principal components analysis would summarize covq ing patterns among variables.

'This paper focuses on the recurring problem of adjusting for a covaiate when regression slopes differ (e.g., Wittle and Fitzsirnons 1983; Johnson et al. 1987; Allad and Stokes 1989) by (I) illustrating the implications of such adjustments on analytical results and subsequent data interpretation, a d (2) demonstrating an alternative multivariate approach to this problem. We use a fish-and-mercury example although this is a common problem relevant to physiology, rnorphometslcs, and many other subdisciplines (e.g., see Reist 1985; Packard and Boardman 1987, 1988). To demonstrate the utility of our multivariate approach, we malyse data from 10 lake trout populations from each of three regions in Ontario to search for large-scale regional patterns as possible evidence of atmospheric deposition (i.e., spatial autocorrelation) (e.g., Jumars et al. 1977; Sokal and Bden 1978; Mackas 1984; Fortin et al. 1989).

Future studies should contrast these multivariate results with lake physical and chemical features to search for potential cause-and-effect relationships (as in McMuW et al- 1989; Cope et al. 1990; Wiener et al. 1990). For example, spatial patterns in the multivariate results could be compsed with patterns in the physical and chemical variables using Mantel tests (e.g., Jackson

md Harvey 1989; Legendre and Fortin 1989) and Procmstes analysis (e.g., Digby and Kemptsn 1987, p. 112-123; Rising and Somers 1989; Rohlf and Slice 1990). Similarly, Iscdized pattems could be explored with partial Mantel tests that first remove large-scale regional patterns (e.g., Hubert 1985; Manly 1986; Srnouse et 31. 1986; Bocquet-Appel and Sokal 1989).

Methods

To explore this problem with a balanced design focusing on three geographic regions in Ontario, we selected 30 of the 91 regressions for lake trout populations that were described by McMu&ry et al. (1989). Ten lakes were chosen from north- western Ontario (i.e., with latitude >48" md longitude >89"), 10 from the Algoma area (i.e., 46-48" latitude and 82-85" longitude), and 10 from the Algonquin area (i.e., 4446" latitude and 77-81" longitude) (see Table 1). Individual lakes were selected if at least 20 fish were sampled. Lakes with samples of at least 15 and 13 fish, respectively, were included to obtain 10 lakes from northwestern Ontario and the Algorna region.

Raw data for fish size and mercury concentration were sum- marized with the mean, standard deviation, minimum, and maximum to provide eight univariate summary variables (see Tablie I). Regressions of the logakith s f mercury concentration in dorsal muscle and fish length (as in McMuary et al. 1989) generated three bivkate statistics for each population (i.e., slope, intercept, and correlation). One-way ANOVAs were used to contrast these 11 variables from the three regions to assess geographic differences (SAS Institute Inc. 1988). A posteriori differences were identified with the Heast-significant-difference test.

The impact of subjectively choosing an adjusted size was explored by calculating predicted mercury concentration at fish lengths of 25,44, 100, and 200 cm using individual population regressions, within-group regressions, and the common reges- sion across all populations (i.e., approximately 0.5, 1, 2, and 4 times the value used by McMudry et alal. 1989). The resultant predicted mercury concentrations were conelated with the orig- inal population parameters using Speaman rank correlations.

Multivariate differences were assessed with a MMOVA and canonical variates analysis (CVA) @AS Institute Inc. 1988). We then summarized patterns among the lakes with an ordination. The eight univaiate and thee bfvarjlaie summary statistics were each standardized to a 0-1 range to weight each variable equally without changing the distribution of the values. A Euclidean distance matrix was then calculated between the 30 lakes using these 11 standardized variables. This distance matrix was rotated to the first five principal coordinate axes with principal coor- dinates analysis (PCoA) (Gower 1966). Loadings of the 11 variables on these new axes were assessed with Speaman rank correlations because of nonnomality in the data. S p e m a n rank correlations were also calculated between the predicted-mercury variables and the PCoA axes.

To test for spatial autocomelation, the Euclidean distance matrix based on the first five principal coordinates was compared with an interlake geographic distance matrix using a Mantel test and associated comelogram (e.g., see Sokal 1979; Legendre and Troussellier 198 8; Bocquet- Appel md Sokal 1989; Legendre and Fortin 1989). We only used the first five PCoA axes to omit unccsrrelated variation embedded in the entire data set (i.e., random noise) (see Gauch 1982). Both Peason and S p e m m correlations were evaluated in the Mantel tests (and associated

Can. .I. Fish. Aquat. Sci., I'u1.50,l99.3

TABLE 1. Location, sample sizes, and univxiate and bivxiate summary statistics for 30 lake trout populations from thee regions in Ontario.

Region Regression Fish length (cm) Mercury concn. (pg g-l)

Lake name Lat., long. N 2 Slope Intercept Mean SD Min. Max. Mean SD Min. Max.

Northwestern Ontario Bending Greenwater Pickerel Snook Basswood Clearwater Paguchi Quetico Sandy Beach Trout

Algoma Bums Esten Wawa Cobre Como Flack Lac Aux Sables Matinenda North Hubert Saymo

Algonquin Bark Big Porcupine B onnechere Bum's camp Drag Fletcher Gull Hdiburton Kennisis

comelograms) using two-tailed probabilities and 10 000 ran- domizations (as recommended by Jackson and Somers 1989; also see Legendre and Fortin 1989).

Results

One-way ANOVAs of the raw data csnfimed concerns expressed by McMuary et al. (1989). The mean length of lake trout collected from northwestern Ontario Bakes was signifi- cantly greater than the mean for Iake trout collected fiom Algoma and Algonquin lakes (F = 5.29 with 2,27 df, P < 0.012, r" 0.282). In addition, the maximum length of lake trout differed among the three areas (F = 4.77 with 2,27 df, P < 0.017, r" 0.262). Maximum fish sizes from northwestern Ontario lakes were significantly larger than the Algonquin samples, and the Algoma fish were intermediate. None of the other summary statistics differed significantly among the regions (P > 0.1).

Regressions of the logarithm of mercury concentration and fish length displayed significantly different slopes across the 30 populations (ANCOVA, F = 4.55 with 29, 420 df. P < 0.801; with a common within-group regression slope of 0.0201). Sig- nificant heterogeneity of variance was also evident (F,n, = 8.93 with 15, 18 df, B < 0.001). In addition, regression slopes were significantly different within regions (i.e., northwestern Ontario:

F = 2.33 with 9, 179 df, P < 0.025, see Fig. %A; F,,, = 4.25 with 22,16 df, P < 0.4305; Algoma: F = 7.93 with 9,214 df, P < 0.001; F,, = 6.41 with 15, 11 df, P < 0.005; ABgonquin: F = 2.69 with 9, 225 df, P < 8.01; F,, = 7.18 with 33? 18 df, P < 0.W.5). The common, within-group regression slope was largest for the set of Algonquin lakes (0.0222) and decreased for the Algoma (0.0209) and northwestern Ontaris lakes (0.0173).

Predicted mercury concentrations at lengths of 25, 44, 100, and 200 cm showed no significant differences among the regions (ANOVA, P > 0.2). Mercury values at 25 and 44 crn that were estimated with the individual regressions were significantly correlated with mean fish length, mean mercury concentration, and the regression intercept (P < 0.05) (Table 2). Mercury csn- centrations at a 25-cm length were also correlated with the regression slope. Correlations with mean fish length suggest that the adjusted values for fish 25 or 44 cm in length based on the individual regressions failed to remove the influence of fish size. In contrast, predicted mercury concentrations at 108 and 200 cm were correlated with the regression slope m d correlatisn, but not mean size. Adjusted mercury at 100 cm was also correlated with mean mercury concentration whereas the 200-cm mercury concentration was correlated with the regression intercept.

These findings changed when using the common-slope adjust- ment or the comon, within-groups regression adjustment

Can. J . Fish. Aqu~r . Sci., Vo1.50, 6993

TABLE 2. Speaman rank con-elations ( ~ 1 0 0 0 ) between common within-group-adjusted and individual-regression-ad-iusted mercury concentrations and univariate, bivariate, and multivariate summaries for lake trout populations from 30 Ontario lakes (rs = 0.375 at B < 0.05, 0,483 at P < 0.01, 0.598 at B < 0.001).

Regression Speamm rank csmelation ( ~ 1 0 0 0 ) adjustment - --

and associated Mean Mean Regression Principal coordinate axis size for [Wg] fish Hg prediction length concn. Slope Intercept Comelation I 11 1 IV V

[Hg] at 25 srm [Hg] at 44 cm [Hg] at 100 cm [Hg] at 200 crn

[Hg] at 25 cm [Hg] at 44 cm [Hg] at 100 can [Hg] at 200 cm

Commoiz slope across groups

bncBE'vidua1 reglassiorz slope an66 inferccg f

TABLE 3. Speaman rank cowelations ( ~ 1 0 0 0 ) betmen univariate and bivariate summary statistics and the first five principal coordinate axes for lake trout populations from 30 Ontario lakes (r, = 0.375 at P P 00.5, 0.483 at P < 0.01, 0.598 at B < 0.001). Bashes within the body of the table represent cornlatioras between a variable anel itself.

Speaman rank correlation ( ~ 1 0 0 0 ) Univaiate ---

and bivaiate Mean Mean Regression Principal coordinate axis s u m m q fish Hg ---

statistics length concn. Slope Intercept Correlation I 11 III IV V

Mean length Length SD Min. length Max. length Mean [HgJ CHgl SD Min [Hg] Max [Hg] Slope Intercept Correlation

(Table 2). Correlations between predicted merckary concen- tration using the within-groups adjustment and mean fish length or mean mercury concentration resembled con-elations based on the iwdividual-regression adjustment. However, correlations with the regression statistics differed (i.e., depending on the adjusted size), with no significant csrrelatioras with the regression slope or associated correlation coefficient. Clearly, adjusting &ta to a common size affects subsequent results and interpretations when regression slopes differ-

A PCoA sf the univariate and bivariate statistics summarized 95 % of the standardized variation in the first five axes (Table 3). The first axis accounted for 58% of the variation and included seven of the univariate variables (excluding the standard

2392

deviation of fish length) and the regression intercept. The second PCaA axis contrasted the standard deviation sf length, plus the regression slope and comelation with minimum fish length and intercept (i.e., 24.4% of the variation).

One-way AWOVAs of the five sets of K o A scores indicated that the three regions differed on the first axis (F = 2.64 with 2, 27 df, P < 0.09, r2 = 0.164) and differed significantly on the fourth axis ( F = 5.98 with 2, 27 df9 B < 0.008, r" 0.307). This fourth axis contrasted the maximum and standard devi- ation of fish length with the regression slope (Tcable 3; i.e., 7.9% ~f the variation). The Algonquin lakes had greater mean scores compared with the Algsrna and northwestern Ontario lakes.

Cun. .I. Fish, Acjuar. Sci., kbf. 50,1993

PCOA 1 (50%) length Ha

Fro. 2. Scatterplot of the first and secsrmd PCoA axes derived from the analysis sf Euclidean distances among 30 lakes based on eight mivariate and three bivwiate summary statistics from data on Iake trout length 'and tissue-mercury soncentratiesns.

A MANOVA confirmed the univariate findings by revealing significant differences among the regions using the 1 1 variables (Wilk's lambda = 0.165, F = 2.25 with 22, 34 df, P < 0.01'7: Roy's Greatest Root = 2.98, F = 4.87 with 11, 18 df, P < 0.002). The CVA (not shown) emphasized that the multivariate differ- ences arose because fish from northwestern Ontario were lager than the Algsma or Algonquin samples. A MANOVA using the five PCoA axes was also significant (Wilk's lambda = 0.490, F = 1.97 with l0,46 df, B < 8.059; Roy's Greatest Root = 0.868, F = 4.16 with 5, 24 df, B < 0.008), emphasizing that the stmdardization and the choice to use only the information in the first five axes reduced the multivariate separation among the three regions.

The thee sets of lakes overlap in the space defined by the first two PCsA axes (Fig. 2). Most of the northwestern Ontario lakes lie on the right side of the ordination, although this space is shared with four Algoma and two Algonquin lakes. These popu- lations are characterized by larger fish with higher mercury csn- centrations and larger regression intercepts (Table 3). The remaining Algorna and Algonquin populations are intermixed, suggesting similarities m s n g these populations with respect to the mercury a d fish-length data.

The MANOVA based om the ordination revealed that popu- lations of lake trout from the northwestern Ontario lakes show modest separation from the other populations (see Fig. 2). The overlap among lakes from the various regions suggests that factors contributing to differences in tissue mercury vary within a region (e.g., mean and maximum fish length). However, this observation does not discount a local-effects hypothesis that populations from nearby lakes (e-g., within the same watershed) exhibit similar mercury fish-size relationships.

The Mantel test indicated a significant correlation between distances among lakes on the first five PCoA axes and the geographic distances separating those same lakes ( r - = 0.1 16,

-0.2

0 206 400 606 800 24306

DISTANCE (km)

FIG. 3. Mantel test conelogram contrasting interlake distances from the first five PCsA axes and geographical distances. Overall Petaffson and Spearman correlations are presented in the legend whereas class values are plotted (along with probabilities when less than 0. 1). Class sample sizes are 115,70,66,64 and 120, respectively. Distances represent class marks, not midpoints,

P < 0.017; r, = 0.155, P < 0.001). The larger rank correlation suggests that the relationship between the two distance matrices is curvilinaear, with differing local and regional effects.

A conelogram showed that positive spatial autoconelatian exists mong the nearest lakes, as well as the most distant lakes (Fig. 3). This observation suggests that mercury fish-size

Ca~a. J . Fish. Aquat. Sci., \+I/. 50,1993 2393

relationships in lake trout populations in neighbouring lakes are similar (r, = 0.155, P < 0.098; is. , supporting the local-effects hypothesis). However, the strongest association existed between the most distant lakes (i.e., between regions; r, = 0.243, P < 0.808), indicating significant large-scale regional patterns (i.e., the weak geographic clusters &sewed in Pig. 2). The shape of this correlogrm is characteristic of a surface with a central peak or depression (e.g., see Eegendre and Fortin 1989), although this pattern incorporates covariation associated with differences in fish length (i.e., length-related infomation in PCoA axis I that varies across regions; see Table 3).

A Mantel test and associated comelogram based on a distance matrix using the second through fifth BCoA axes omits a multivariate analogue of these length-correlated effects (e.g., see Somers 1989) and examines the spatial pattern of this four-dimensional, configuration (Lee, 46.1% of the standardized variation in the original 12 variables). The overall Mantel test was not significant (r = 0.023, P < 0.63'9; pq, = 0.05 1, B < 0.299), emphasizing that the large-scale regional pattern was mainly based on the fish-size information in the first PCoA axis (see Table 3). The comelogram (not shown) was similar to the one based on all five axes (Fig. 3), with both srnall-scale (i.e.. <200 h: I-, = 0.164, P < 0.085) and large-scale spatial auto- correlation approaching significance (600 h: P., = 0.2 1 3, B < 0.086; 800 h: rs = 0-232, P < 0.068; >800 Ham: p., = 0.151, P < Oe098). The MANOVA using these four ordination axes provided similar results (Wilk's lambda = 0.614, F = 1.65 with 8,48 df, P < 0.135; Roy's Greatest Root = 0.625, F = 3.90 with 4,25 df, B < 0.014), given that Roy's Greatest Root has greater power to detect a gradient than Wi&'s lambda (Clarke and Green 1988). These findings suggest that both local and regional pat- terns remained once the dominant fish-size effects were removed (i.e., patterns within areas, as well as among areas).

McMufiv et al. (1989) assumed that their standardized mercury concentratisn represented a size-adjusted concentration for each lake trout popu1atior.a. Our analysis of a subset of their data substantiates this assumption (see Table 2). However. om size-adjusted data were also correlated with mean fish length and mean mercury concentration, suggesting that the standardization failed to remove the confounding influence of fish size. Had McMurtry et al. (1989) chosen an adjusted length of 25, 108. or 208 cm (recognizing that these lengths would be extrapolations beyond the data), their results would have probably differed. The choice of an alternative regression adjustment (e.g.. the within- groups regression) would also affect their results.

Given these inconsistencies with size adjustments when slopes differ, we propose that univariate and bivdate s u m m v statis- tics should be analysed with multivariate methods as m alter- native to the use of a single adjusted value. Similar approaches are common in rnorphornetrics (e.g., Atckley and Rutledge 1980), especially when using Fourier analysis methods (e.g., Rohlf and Archie 1984: Person et al. 1985). An ordination of univaiate and regression statistics provides less ambiguous results and summaizes covarying patterns into uncorrelated axes for comphson with other suites of variables.

Since regressions are influenced by sample sizes, plus the standard deviations md ranges of the variables, the discovery of clusters of populations or outliers in the resultant ordination will assist in resolving differences among populations. The ordi-

nation in this example revealed patterns reflecting geographical proximity (i.e., spatial autoconelation), perhaps due to both local watershed or lake basin geology, and large-scale patterns that might be attributable to atmospheric inputs (e.g., Rada et al. 1989; Mierle 1990). In addition, prey type can influence contm- inant accumulation (e.g., MacCH-immon et al. 1983; Mathers and Jokansen 1985; Cope et al. 1990), so biogeographical patterns associated with the presence or absence of particular prey species could also contribute to observed groupings of lakes (Rasmussen et al. 1990).

To extend the original study of McMuPtry et al. (1989), the resultant ordination axes could be compared with patterns from ordinations of physical or chemical variables using a Mantel test or Procmstes analysis (e.g., Bigby and Kernptsn 1987, p. 112-123; Jackson and H m e y 1989). Such methods have fewer restrictive assumptions than multiple regression, espe- cially stepwise procedures (Rechow et al. 1987; James and McCulloch 1990).

Perhaps one drawback to this approach is the difficulty in resolving or interpreting multivariate axes (James and McCulloch 1990), although the reification of such axes is now widely accepted (Pimentel 19'79, p. 7; Reyment 1990). Given that these types of studies are inherently multivariate (i.e., based on several response variables), single summary indices (such as the adjusted mercury concentration described herein) are inappropriate a d can contribute to misleading results and interpretations (e.g., see Jackson et al. 1990).

To s u m m ~ z e , we caution against the routine use of individual or common within-group regressions to adjust data to remove the influence of a predictor variable when regression slopes differ. These types of adjustments c m produce misleading results reflecting differences in regression slopes rather than intercepts, or retaining information correlated with the original size variable. As an alternative, we illustrate that multivariate pro- cedures can be used to examine univariate and bivaiate s u m m q statistics t s resolve and interpret patterns with less information loss. In addition, the resultant ordination axes can be used to evaluate correlations with various environmental fac- tors, including geographic location (i.e., spatial autocomelation).

Acknowledgments

We thank R.W. Bilyea, R.L. France, D.L. Fuller, H.H. Harvey, M.A. Keenleyside, M.J. McMuhaay, M.S. Ridgway, %:A. Scheider, R. Snell, D.L. Wales, and C.D. Wren for discussions and helpful suggestions. H.H. Harvey supervised this work. D.L. Wales, from the Fisheries Branch s f the Ontario Ministry of Natural Resources, kindly provided the raw data for the mercury fisk-size relationships. Portions of this study were funded by an NSEQC scholarship to D.A.J. and an NSERC operating grant to H.H. Marvey.

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Call. J . Fish. Aquar. Sci., V01.50, 1993